Long-range dependence in third order and bispectrum singularity

In this paper the third order long-range dependence (LRD) is de…ned in terms of the bispectrum and third order cumulants (bicovariances). Two particular non-Gaussian processes with second order LRD are considered together with their bispectra and bicovariances. AMS 2000 Subject Classi…cations: Primary 62M10, 37M10; secondary 62M15, 91B70. Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays. 1 Introduction One of the …rst applications of the bispectrum was in oceanography, where statistical dependence arises from wave triad interactions, which are the lowest-order nonlinearity in weakly nonlinear wave …elds ([12]). The bispectrum has been used to examine nonlinear interactions among various measurements for instance the Fourier components of turbulence. Bispectral analysis has also been applied to boundary layer dynamics, acoustics ([9]), seismics ([16]), plasma physics, economics, and neurology ([4]). The rigorous theoretical study not only of bispectrum but of higher order spectra as well has been suggested by Kolmogorov and Tukey independently, some pioneering works are [6], [19], [18]. The use of bispectrum includes …elds such as testing Gaussianity and linearity ([24], [14], [28], [26]), deconvolution and phase reconstruction for non-Gaussian linear processes ([20]), re…ned parameter estimation ([7]) and so on. Some possible singularity of the bispectrum has been mentioned in context of music identi…cation ([8]) and turbulence ([13]). The object of this paper is to de…ne the long-range dependence (LRD) in third order for nonGaussian time series. In paper [27] the de…nition of LRD has been based on the particular form of the bispectrum and bicovariances, this time we use the properties of those on the boundary of the principal domains and arrive the particular form as a consequence. It is shown that this de…nition is valid for the non-Gaussian fractional noise and Rosenblatt processes. The LRD of a …ltered stationary process is also considered as a by-product of the frequency domain method applied here intensively. 2 Bispectrum and Bicovariances We consider a stationary in third order time series Xt, t = 0; 1; 2; : : : , i.e. not only the covariance function of Xt is invariant under the time shift but the third order cumulants as well. More precisely, Cum(Xt+s1 ; Xt+s2 ; Xt) = C3 (s1:2) ; s1; s2 = 0; 1; 2; : : : : where s1:2 = (s1; s2), for short. These third order cumulants are called bicovariances as well, see [17], [15]. An easy consequence of this de…nition is that the following properties are ful…lled C3 (s1:2) = C3 (s2; s1) = C3 ( s1; s2 s1) :